How to control the flow pattern of the noise texture node?
In this image you're using UV space and then scaling the X axis to 0, so that's why your noise texture is coming out like a wave texture.  I'll go through what my solution might look like.  This is UV space so you can either just go and use proportional editing to move the UVs of the faces, or we can do it with nodes.
Here's our sample scene, it's just a plane and I've put a wave texture on it.

We'll use your distance from the UV origin on one axis to move the UV space along the other axis.  We'll use a separate XYZ and a combine XYZ node.

And then you can insert any kind of math in there to control the rate at which you are shifting the UV space over, or create any manner of effect.